SALES FORECASTING IN A MECHANICAL COMPONENT MANUFACTURER:
COMPARISON BETWEEN MONTE CARLO SIMULATION AND TIME SERIES ANALYSIS
Kevin William Matos Paixão
Instituto Federal de São Paulo  Campus Suzano, Brazil
Email: kevinwilliamp@gmail.com
Adriano Maniçoba da Silva
Instituto Federal de São Paulo  Campus Suzano, Brazil
Email: adrianoms@ifsp.edu.br
Submission: 01/31/2019
Accept: 02/10/2019
ABSTRACT
Organizations
today are required to be prepared for future situations. This preparation can
generate a significant competitive advantage. In order to maximize benefits,
several companies are investing more in techniques that simulate a future
scenario and enable more precise and assertive decision making. Among these
techniques are the sales forecasting methods. The comparison between the known
techniques is an important factor to increase the assertiveness of the
forecast. The objective of this study was to compare the sales forecast results
of a mechanical components manufacturing company obtained through five
different techniques, divided into two groups, the first one, which uses the
fundamentals of the time series, and the second one is the Monte Carlo
simulation. The following prediction methods were compared: moving average,
weighted moving average, least squares, holt winter and Monte Carlo simulation.
The results indicated that the methods that obtained the best performance were
the moving average and the weighted moving average attaining 94% accuracy.
Keywords: sales forecast, Monte Carlo
simulation, mechanical components
1. INTRODUCTION
Technological evolution and management techniques and
methods development will make the market competition fierce. The simple
exchange system of products and services that was once the basic essence of the
market, has given way to a crowded and contested environment where only those
companies that differentiate themselves from competitors will stand out
(MCKENNA, 2005).
Achieving this advantage is a major challenge, since
the management sectors use strategies and conduct planning based on market
analysis as well as competitors and customers. All this effort is carried out
in order to increase the company’s competitiveness.
A wellstructured company should not only concern
itself with current trends but make decisions aiming at medium and longterm
results. However, in order to achieve these results and become more
competitive, several companies use techniques that can estimate a foreseeable
future. These techniques are known as demand forecasting.
For Haack and Rodrigues (2018), predicting demand
means estimating future market positions and anticipating the consumer's
response to their product. This trick uses two different methods on its
procedure. The first is the qualitative method which consists of a forecast
made by experts based on their own analysis. The second is the quantitative
method, which consists of a numerical estimate generated through historical
analysis. This model can be made through different mathematical procedures.
Within a decisionmaking environment, demand
forecasting with a higher level of accuracy can benefit several sectors of an
organization. According to Werner and Ribeiro (2006), the selection of which
technique a company should use in its forecasts must take into account the
degree of complexity of the multiple manufacturing sectors, thus seeking
greater accuracy.
A systematic search with the terms "sales
forecast" and "forecast demand" was performed in the SPELL
database that resulted in nine articles dealing with these issues. Some papers
compared demand forecasting techniques based on time series. However, no study
compared the efficacy of these techniques with the Monte Carlo simulation.
The purpose of this study was to compare demand
forecasting results with the sales made by an electronic component manufacturer
using time series methods and Monte Carlo simulation.
The work was structured in six sections: introduction,
literature review, methodology, results, discussion and conclusion.
2. LITERATURE REVIEW
2.1.
Demand
Forecasting System
In
order to define the best demand forecasting system to use, Pellegrini and Fogliatto (2001) point out that three basic conditions must
be taken into account: the availability of historical information, the
possibility of transforming information into data, and the assumption patterns
repeating themselves. These are the assumptions used to select a demand
forecasting method.
A
time series, or "historical series" as it is also known, is defined
by a series of information (numerical values) obtained in a regular space of
time. These numbers can be obtained by periodically observing the event under
study or by counting. According to Pellegrini and Fogliatto
(2001), the measurement units of this element and the size of the observed
period have influence in the choice of the mathematical procedure to be used.
According
to Makridakis (1998) and Moreira (2001), a time
series has four known components: The average component, which is the simplest
behavior that a series can have, which occurs when the values of a series
oscillate between constant an average value; Component trend, which has
behavior indicating whether the series will have growth or reduction; Seasonal
component, which is based on the lack or excess demand of certain items always
in very specific periods. Following the reasoning of Morettin
and Toloi (1987) seasonal products can also be called
annual or stationary components; Cyclical component, which, according to
Moreira (2001), are fluctuations of general order and with varying frequency;
and Random component, being an anomaly caused by events of known or unknown
order.
According
to Pellegrini and Fogliatto (2001), the results have
two basic elements that are: Horizon, which can be defined as the temporal
distance that the results allow to visualize; and the second element is an
interval, which defines the frequency at which new forecasts are prepared.
Within
the quantitative methodology of demand forecasting, we can find two distinct
systems of mathematical models, which according to Armstrong (2001) are: System
of causal methods, in their composition causal variables are estimated and are
related to estimates of independent variables; Extrapolation system, whose
principle of operation lies in the assumption of constant patterns and
stationarity of the data (historical series) used in the process, i.e., this
system predicts future patterns through the relation of values existing in
the past.
According
to the reasoning of Lemos (2006) the extrapolation
system is divided into two methods. And these are: Open Mathematical Modeling
Methods, which are characterized by specific methods, which consider
characteristics of each business within the historical series. Although this is
a very accurate method, its use requires high investment to identify specific
components; and Methods with Fixed Mathematical Models, which are simple, low
cost methods, since they have fixed equations that consider the basic factors
of the historical series. This method is recommended for shortterm forecasts
and environments undergoing constant change.
2.2.
Forecasting
methods
2.2.1. Moving
Average (MA)
According
to Tubino (2009), the main characteristic using the
moving average as a mathematical procedure within the forecast of demand is the
combination of high and low values, thus generating a forecast with little
variability. The operation of this mathematical procedure consists of dividing
the sum of values obtained during a given period by the number of periods. Its
mathematical expression is exposed as follows:
Mp= average (forecast)
P1= Period 1
P2= Period 2
P3= Period 3
P4= Period 4
Pn= Possible numerical values
n= Number of values that
were summed
This forecasting method is called the moving average
because, at the end of a period, the actual result obtained in the calculation
is added and the oldest period is extracted. In this way, the forecast remains
within the real trend. According to Davis, Aquilan
and Chase, (2001), this method has low accuracy when dealing with seasonal
products. To solve this problem, it is common to use a variation of this
procedure, called the weighted moving average.
2.2.2. Weighted
Moving Average (WMA)
This
procedure is similar to the simple moving average formula presented above, the
difference being that weights are used to determine the influence of the
periods on the final result.
M_{p}= Average (forecast))
P_{1} = Period 1
P_{2} = Period 2
P_{n} = infinite values
x_{1} = Ponderation
1
x_{2 }= Ponderation
2
x_{n} = possible number of ponderations
In
this model, the principle of replacing the last period with the most recent
one, which has already been completed, remains. Tubino
(2009) states that the disadvantage of this procedure lies in the fact that it
is necessary to have a specialized opinion to propose the weighting values.
2.2.3. Holt
Winter Method
The
Holt winter method, according to Chopra and Meindl
(2003), consists of a system of estimating demand components based on the
already existing ones in the time series. Basically this is also an exponential
smoothing method in which the time series is decomposed into its components,
and they are weighted relative to the time distance. This weighting is applied
in the forecast, maintaining the pattern of the time series components.
The
decomposition can take two different forms; Multiplicative and Additive and
what determines this form is the nature of the demand. To measure the forecasts
through Holt winter the formulas in Table 1 are used:
Table 1:  Holt Winter Formulas
Multiplicative 
Additive 

Level 


Tendency 


Seasonality 


Forecast 


Source: Adapted from Chopra e Meindl (2003)
Where:
s  Length of seasonality
Lt  Series Level
bt  Trend
St  Seasonal component
Fm + t  Forecast for the period m ahead
Yt  Observed value
α, β and γ  Smoothing exponential
parameters, level, trend and seasonality, respectively.
2.2.4. Least
Squares Estimate (LSE)
Helene
(2006) states that the least squares estimate is the most popular method and
one of the most efficient in the treatment of experimental data. Briefly, this
procedure consists in the search for a mathematical function that best fits in
a series of points plotted in a Cartesian plane. The main premise in this
process is the principle of minimizing the square of the sum of the differences
(error) between the estimated value and the observed value.
Figure 1: Line
of representation of points
Source:
The Authors
e_{1 = }Error 1
e_{2 = }Error 2
e_{3 = }Error 3
e_{4 = }Error 4_{}
e_{n}_{ = }Error 5
V_{r}1= Real sales
value on Period 1
V_{r}2 = Real sales
value on Period 2
V_{r}3 = Real sales
value on Period 3
V_{r}4 = Real sales
value on Period 4
V_{r}5 = Real sales
value on Period 5
P = Forecast for period 6
According
to Hair et al. (2009), the linear regression procedure estimates the degree of
association between a dependent variable (sold values) and an independent
variable (periods), and thereby determines a correlation (trend) between them.
2.2.5. Monte
Carlo Simulation
Monte
Carlo Simulation is a mathematical tool of operational research capable of
creating a simulation of stochastic order scenarios, that is, scenarios that
depend on a random and unpredictable variable. This method was named during
World War II during the Manhattan project, making reference to the city of
Monaco known at that time as the capital of gambling. Its principle of
operation is statistical, where through a random sequence of numbers an event
simulation is generated in which the average behavior of the variables is the
ideal estimated solution. With this Fernandes (2005) divides the Monte Carlo
process into 3 phases; First, establish a probability distribution (random
variables of the problem) and correlate them with random numbers that will simulate
the random variable (random number generator); The second, sample again and
again (sampling techniques); The third, calculate the average behavior of the
samples and the standard deviation thus obtaining the ideal estimated solution.
According
to Donatelli and Konrath
(2005) the best application for the Monte Carlo simulation is in mathematical
systems that do not allow an analytical solution due to the unpredictability of
the information, since it is a technique with high effectiveness in random
statistical sampling.
Garcia,
Lustosa and Barros (2010) apply the Monte Carlo
simulation to predict the cost of production of industrial companies, using 28
entities, and generating 5600 simulations for them. Mendes, Silva and Kawamoto Júnior (2016) used Monte Carlo simulation to analyze the
variability in capacity caused by human behavior.
In
addition, Monte Carlo simulation is applied to a process of risk analysis,
where it points out important aspects in the creation of any simulation: the
choice of an adequate level of confidence and the application of empirical
distribution in a coherent way. In addition, it evidences the fact that a
considerable sample must be contained to obtain feasible results.
2.3.
Average
Absolute Percentage Error
To
evaluate the performance of a forecasting technique, it is necessary to compare
the predicted data with reality. In agreement with Heizer
and Render (2004), this mathematical procedure consists of the average of the
absolute difference between the predicted values and the actual values
expressed as a percentage of the values reached. In order to obtain this
result it is necessary to go through two stages.
1st
stage: Obtaining the individual percentage error (PE)

EP = Percentual Error
Vr1 = Real
value obtained in the first period in which it was intended to predict
P1=
previsão para Vr1

EMP = Average Percent Error
 EP1
 = Percentage error module obtained in the first forecasting period
 EP2
 = Percentage error module obtained in the second forecast period
 EPn  = percentage error module obtained in the numerous
forecast periods
n =
amount of PE used in the sum
3. METHODOLOGY
This
study used the case study methodology applied at a company in the Alto Tietê region of the state of São Paulo, located in the city
of Suzano. Its main product is a specific mechanical
component. Data on components sold per month were obtained. The sample
collected represents 96 months, from September 2011 to August 2018, totaling
seven years. The forecast models were analyzed with 84 months and the last 12
months were used to analyze the prediction performance. The collected data used
can be visualized in Table 2.
Table 2: Data
collected
2010/11 
2011/12 
2012/13 
2013/14 
2014/15 
2015/16 
2016/17 

September 
3.627.157 
3.891.218 
3.329.907 
3.602.552 
3.112.388 
3.073.420 
3.498.121 
October 
3.460.229 
3.772.815 
3.411.403 
3.654.776 
3.743.145 
3.299.349 
3.281.806 
November 
3.551.704 
3.589.642 
3.573.171 
3.378.208 
3.384.299 
3.260.945 
3.337.972 
December 
2.582.564 
2.782.598 
2.438.752 
2.526.394 
2.798.412 
2.335.088 
2.990.215 
January 
3.205.249 
3.259.099 
2.987.974 
3.206.115 
2.807.163 
3.118.495 
3.467.309 
February 
3.278.786 
3.251.362 
3.515.425 
3.009.833 
3.113.183 
2.921.120 
3.246.857 
March 
4.212.786 
3.293.631 
3.296.673 
3.056.192 
3.682.002 
3.532.763 
3.717.665 
April 
3.686.997 
3.458.653 
3.743.282 
3.239.210 
3.307.913 
3.086.425 
2.963.617 
May 
3.472.292 
3.170.491 
3.677.122 
3.301.557 
3.250.123 
3.377.096 
3.454.247 
June 
3.514.847 
2.824.897 
3.385.928 
2.577.480 
3.035.108 
3.184.060 
2.690.163 
July 
3.675.996 
3.369.053 
3.521.605 
3.288.554 
3.205.254 
3.488.280 
3.783.167 
August 
3.795.276 
3.858.877 
3.693.130 
3.163.077 
2.988.775 
3.417.848 
3.566.406 
Thus,
five forecasts were obtained each month, all of which were analyzed and
compared with the previous year's sales results, using the Average Absolute
Error (AAE) as a reference.
In
the Holt Winter model the following specific steps were followed:
·
1st Step  Demand deseasonalization
·
2º Step Estimate of level and trend with linear
regression applied to seasonally adjusted demand as a function of the period;
·
3º Step Estimation of seasonality factors;
·
4º Step  Application of the estimation in the demands
to obtain the forecast;
·
In the Monte Carlo method, the following steps were
followed:
·
1st Step –Testing adherence to the demand seasonal
demand adjusted and without a complete trend (creation of the PDF).
·
2nd Step  Calculation of the number of necessary
simulations.
·
3rd Step  Generation of 50240 simulations for
seasonally adjusted demand.
·
4º Step Application of the seasonality and trend
factor of each month of the year in the average of all the values generated
in the simulations
In
order to perform the cited procedures, computational resources of the Arena and
Calc software were used. The execution of this
process served the basic purpose of scientific research that according to Severino (2017) consists in the knowledge of an object in
its primary sources and foundations. The term "object" mentioned
above can be attributed all mathematical content and the mechanics of data
analysis that were explored in this article.
4. RESULTS
4.1.
Time
Series
The
results of time series forecasting methods can be seen in Table 3.

Forecasting
Method 
MA 
WMA 
Holt
Wilter 
LSE 
September 
3.447.823 
3.412.556 
3.234.951 
3.127.791 
October 
3.517.646 
3.459.259 
3.302.390 
3.353.295 
November 
3.439.420 
3.406.868 
3.228.014 
3.226.926 
December 
2.636.289 
2.675.366 
2.477.883 
2.734.517 
January 
3.150.201 
3.193.963 
2.959.670 
3.196.509 
February 
3.190.938 
3.182.030 
2.994.540 
3.025.436 
March 
3.541.673 
3.561.694 
3.324.966 
3.452.849 
April 
3.355.157 
3.262.376 
3.143.648 
2.876.590 
May 
3.386.133 
3.396.245 
3.179.335 
3.376.429 
June 
3.030.355 
2.986.956 
2.840.817 
2.729.420 
July 
3.475.987 
3.525.616 
3.263.920 
3.510.789 
August 
3.497.627 
3.503.725 
3.278.720 
3.172.909 
Average 
3.305.771 
3.297.221 
3.102.404 
3.148.622 
Source:
The authors
In the execution of the mathematical procedure of the
weighted moving average, according to the content exposed in the bibliographic
review, an executive from the company was consulted, to collect the data to
carry out the weighting of the time series. Table 4 presents the weighting and
its justifications:
Table 4: Weights
used
Year 
Weighing 
Justification 
Influence on forecast 
1 
2 
Promising market, growing economy for the old
period. 
8% 
2 
3 
Increasing automotive market, and stable
postelection period. 
12% 
3 
3 
IPI reduction policies kept the car market stable. 
12% 
4 
2 
Fall in growth and retraction in the industrial
market. 
8% 
5 
3 
Close period and high drop in the automotive and
industrial market. 
12% 
6 
5 
Economic retraction in the country driven by
political crisis. 
20% 
7 
7 
Closest period with recessive economy and market for
investment 
28% 
Source:
The authors
Also obtained through the least squares estimate were
dispersion diagrams with trend lines. Figure 2 presents two examples.
Figure 2: Trends
dispersion diagrams
Source:
The authors
4.2.
Monte
Carlo Simulation
4.2.1. Adherence
test
The
adherence test was performed on the Input Analyzer module of the Arena
software. With the 84 observations representing the sales in the last 7 years
(Table 2) the tests performed by the software were the ChiSquare and
KolmogorovSmirnov tests, both of which showed a pvalue greater than 0.05
which guaranteed good adherence to the tested distribution. After this, a
simple frequency distribution histogram was created, it contained 10 classes
whose interval began in the value 2,335,088 and finished in the value
4,212,786, each interval had a amplitude in the value of 208.633 which resulted
in a maximum frequency of 24 entities and a minimum of 1 entity. The PDF
generated through this distribution generated a normal curve in the bell
format.
4.2.2. Number
of simulations required
In order to calculate a number of necessary
simulations based on a percentage of error, Harrel et
al. (2004) propose the following equation:

N '=
is the number of replications
S =
Is the standard deviation of the data collected
X =
Is the average of the data collected
re =
It is the percentage error defined by the user
To
perform the simulation, the data from Table 5 were used to calculate the number
of replications:
Table 5: Data
for Replications calculations
S 
X 
Re 
N’ 
350096,53 
3.305.771 
9% 
50240 
Source: The authors
4.2.3. Histogram
and confidence interval
With the generated PDF and the number of replications defined, the
simulations were generated obtaining the histogram represented in Figure 3.
Table 6: Sample
Size
Sample Size 
50240 
Maximum 
4663990 
Mimimum 
1855230 
Amplitude 
125339,10714 
No. of intervals 
224 
Average 
3251420, 263 
Pattern deviation 
837452, 3166 
Source: The authors
Figure 3:
Simulation histogram
Source: The authors
With this, we can calculate confidence interval, presented in Table 7,
using the following formulas:
Table 7:
Confidence interval
Upper Limit 
Lower Limit 
3258743,293 
3244097,232 
Source: The authors
4.2.4. Monte
Carlo simulation results
Using
the methodology presented, the results of Monte Carlo simulation were presented
in Table 8.
Table 8: Results
of Monte Carlo simulation
Forecasting
Method 
Monte
Carlo Simulation 
September 
3.281.542 
October 
3.174.062 
November 
3.527.835 
December 
3.340.486 
January 
3.380.639 
February 
3.024.965 
March 
3.478.119 
April 
3.498.931 
May 
3.452.617 
June 
3.528.091 
July 
3.452.605 
August 
2.652.435 
Average 
3.316.027 
Sourche: The authors
4.3.
Analysis
and comparison with the last period
With the result of the last period it was possible to compare the
methods as can be seen in Table 9.
Table 9:
Comparison of forecasting methods
Forecasting
Method 
September 
October 
November 
December 
January 
February 
MA 
3.447.823 
3.517.646 
3.439.420 
2.636.289 
3.150.201 
3.190.938 
WMA 
3.412.556 
3.459.259 
3.406.868 
2.675.366 
3.193.963 
3.182.030 
Holt
Wilter 
3.234.951 
3.302.390 
3.228.014 
2.477.883 
2.959.670 
2.994.540 
LSE 
3.127.791 
3.353.295 
3.226.926 
2.734.517 
3.196.509 
3.025.436 
Monte
Carlo Simulation 
3.281.542 
3.174.062 
3.527.835 
3.340.486 
3.380.639 
3.024.965 
8th year
oficial result 
3.360.811 
3.371.101 
3.476.368 
2.741.386 
3.533.791 
3.701.944 
Error analysis 

MA Error 
87.012 
146.545 
36.948 
105.097 
383.590 
511.006 
WMA Error 
51.745 
88.158 
69.500 
66.020 
339.828 
519.914 
Holt
Wilter Error 
125.860 
68.711 
248.354 
263.503 
574.121 
707.404 
LSE Error 
233.020 
17.806 
249.442 
6.869 
337.282 
676.508 
Monte Carlo
Error 
79.269 
197.039 
51.467 
599.100 
153.152 
676.979 
Absolute
Error analysis 

MA
Absolute Error 
87.012 
146.545 
36.948 
105.097 
383.590 
511.006 
WMA
Absolute Error 
51.745 
88.158 
69.500 
66.020 
339.828 
519.914 
Holt
Winter Absolute Error 
125.860 
68.711 
248.354 
263.503 
574.121 
707.404 
LSE
Absolute Error 
233.020 
17.806 
249.442 
6.869 
337.282 
676.508 
Monte
Carlo Absolute Error 
79.269 
197.039 
51.467 
599.100 
153.152 
676.979 
Percentual
error analysis 

EMP MA 
3% 
4% 
1% 
4% 
11% 
14% 
EMP WMA 
2% 
3% 
2% 
2% 
10% 
14% 
EMP Holt
Winter 
4% 
2% 
7% 
10% 
16% 
19% 
EMP LSE 
7% 
1% 
7% 
0% 
10% 
18% 
EMP Monte
Carlo 
2% 
6% 
1% 
22% 
4% 
18% 
Absolute
EMP Error analysis 

Absolute
EMP Error MA 
3% 
4% 
1% 
4% 
11% 
14% 
Absolute
EMP Error WMA 
2% 
3% 
2% 
2% 
10% 
14% 
Absolute EMP Error Holt Winter 
4% 
2% 
7% 
10% 
16% 
19% 
Absolute
EMP Error MMQ 
7% 
1% 
7% 
0% 
10% 
18% 
Absolute
EMP Error Monte Carlo 
2% 
6% 
1% 
22% 
4% 
18% 
5. DISCUSSION
Analyzing
the average value of the errors obtained in the five methods, it is verified
that the Monte Carlo Method reaches the lowest value, "77.397" units,
bringing this value in percentage form, and in comparison with the official
result, this error equals 1% . However, this analysis does not show the real
precision needed, and may negatively influence the company's decisionmaking.
This
low percentage of error was reached due to the consonances between the positive
(more than real) and negative (less than real) differences between the expected
and achieved results, i.e. there is a large amount that was predicted lower
than the real. However, in other months, this quantity is supplied by a high
value predicted in the rise, which mathematically balances the result.
For
this reason, the absolute error analysis (using only the error value modules)
was applied, so that the value of "212.334" units obtained by the
method of Weighted Moving Average. This value, when taken to percentage unit
reaches the house of 6%. This same percentage of error was reached by the
Moving Average method.
Regarding
the Holt Winter and the Least Squares Estimate methods, the average absolute
error rate reached was the same as the Monte Carlo, 9%, although the monthly
error values were different.
These
results contradict Davis, Aquilan and Chase, (2001)
statement, mentioned in the topic of bibliographic review. They claim that the
mobile average method has low accuracy when applied to products that have a
seasonal component.
Another
objection also contradicted is that of Slack (2013), which takes as a negative
point the fact that the weighted moving average method is more influenced by
the most recent period, in the opinion of the authors of this work this
characteristic is positive, because through it the result is the real scenario
experienced by the company.
Another
question that can be raised is in relation to the results found by Garcia, Lustosa and Barros
(2010) and Matias (2006), since both works had smaller amounts of observations
and simulations, which raises doubts if the application of other methodologies
would not be more accurate and with more significant results.
On
the other hand, the argument of Donatelli and Korath (2005) and Fernandes (2005) is confirmed, since the
biggest monthly error estimated by the Monte Carlo occurred in August with a
lower forecast of "818,516" units. The authors of this work credit
this result to the smaller amount of observations (time series) applied in the
analysis, this confirms the proposed statement that the larger the sample
number the smaller the standard deviation and the more accurate the result.
6.
CONCLUSION
Based
on all this information, it can be considered as the best methods to be applied
in the forecast of demand, specifically in this company, the Weighted Moving
Average and Moving Average methods, because they obtained results with a
smaller average perceptual error obtaining a probability of correctness of 94%.
In
this way, it can be concluded that this article reached the objective of
comparing timeseries methods with Monte Carlo simulation and defining the best
method for application in a mechanical component manufacturer.
It
should be noted that this work is limited in relation to other comparisons and
studies made earlier between different forecasting methods, mainly due to the
limited amount of observations. It is important to mention that this study
contributes with quantitative and qualitative information in the areas of
management, logistics, planning and administration with theoretical content and
veridical results that can be used not only as benchmarking but as cases for
academic studies.
As an
alternative for future work, we suggest new studies comparing different
prediction methods, mainly using the Monte Carlo simulation.
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